Linear Algebra with Applications (3rd Edition)

110.201 Linear Algebra
1st Quiz Solution
February 15, 2005
Problem 1 Given the following system of equations:
3 x + 2 y - 5 z = 1
4 x - y + z = 0
x - z = 2
find all solutions using Gauss-Jordan elimination procedure. Is this an example
of consistent system? Why?
Solution: We use Gauss-Jordan:


3 2 - 5
4 - 1 1
1 0 - 1
1
0
2


We rewrite the matrix as :


1 0 - 1
3 2 - 5
4 - 1 1
2
1
0


- 3 × (I)
- 4 × (I)
then:


1 0 - 1
0 2 - 2
0 - 1 5
2
- 5
- 8


÷ 2
then:
1



1 0 - 1
0 1 - 1
0 - 1 5
2
-
5
2
- 8


+(II)
then:


1 0 - 1
0 1 - 1
0 0 4
2
-
5
2
-
21
2


÷ 4
then:


1 0 - 1
0 1 - 1
0 0 1
2
-
5
2
-
21
8


+(III)
+(III)
then we have:


1 0 0
0 1 0
0 0 1
-
5
8
-
41
8
-
21
8


Therefore, we have the answer :



x = -
5
8
y = -
41
8
z = - 21
8
Problem 2 Find the rank of the following matrix




- 1 3 8 - 2 1
- 1 3 9 - 1 3
1 - 3 - 9 1 - 3
0 0 0 0 2




Solution: We find reduced row-echelon form of this matrix:




- 1 3 8 - 2 1
- 1 3 9 - 1 3
1 - 3 - 9 1 - 3
0 0 0 0 2




× ( - 1)
+(II)
÷ 2
2

then:




1 - 3 - 8 2 - 1
- 1 3 9 - 1 3
0 0 0 0 0
0 0 0 0 1




+(I)
then:




1 - 3 - 8